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Abstract. Least squares collocation method (LSC) is a versatile numerical method for solving boundary value problems for PDE. The present article demonstrates the abilities of LSC to solve various problems -in particular, calculations of bending of isotropic irregular shaped plates and multi-layered anisotropic plates. In order to achieve higher accuracy, new versions of the method utilize high-degree polynomial spaces. The numerical experiments demonstrate high accuracy of the solutions.

Understanding the principles and mechanisms of cell growth coordination in plant tissue remains an outstanding challenge for modern developmental biology. Cell-based modeling is a widely used technique for studying the geometric and topological features of plant tissue morphology during growth. We developed a quasi-one-dimensional model of unidirectional growth of a tissue layer in a linear leaf blade that takes cell autonomous growth mode into account. The model allows for fitting of the visible cell length using the experimental cell length distribution along the longitudinal axis of a wheat leaf epidermis. Additionally, it describes changes in turgor and osmotic pressures for each cell in the growing tissue. Our numerical experiments show that the pressures in the cell change over the cell cycle, and in symplastically growing tissue, they vary from cell to cell and strongly depend on the leaf growing zone to which the cells belong. Therefore, we believe that the mechanical signals generated by pressures are important to consider in simulations of tissue growth as possible targets for molecular genetic regulators of individual cell growth.

This chapter is devoted to modeling the properties of composite materials and structures. Mathematical relations describing the nonlinear elastic three-point bending of isotropic and reinforced beams with account of different strength and stiffness behavior in tension and compression are obtained. An algorithm for numerical solution of corresponding boundary-value problems is proposed and implemented. Results of numerical modeling were compared to acquired data for polymer matrix and structural carbon fiber reinforced plastics. A computational technology for analysis and optimization of composite pressure vessels was developed and presented.

The epidermis of a linear leaf, as in Poaceae, is established by parallel files of cells originating from the leaf base. Their feature is symplastic growth where neighboring cell walls adhere and do not slide along each other. We developed a simple mechanical cell-based model for symplastic growth of linear leaf blade. The challenge is to determine what restrictions on cell size symplastic growth creates compared to the free growing cells. We assume an unidirectional growing cell ensemble starting from a meristem-like layer of generative cells and then generating parallel cell rows from every cell of the initial layer. Each cell is characterized by its growth function, and growth of the whole leaf blade is accompanied by mutual adjustment between all the cells. Cells divide once they have reached a threshold area. A mathematical model and its implementation are proposed for computational simulation of 1D symplastic growth of tissues. The question analyzed is how a cell grows in a plant tissue if there is a mechanism for regulating the growth of an isolated growing cell and the behavior of the cell wall matter is elastoplastic. The results of the simulation of linear leaf blade growth are compared to those for a free-growing cell population.

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This paper is devoted to modeling the properties of composite materials and reinforced composite beams. Mathematical relations describing the nonlinear elastic three-point bending of isotropic and reinforced beams with account of different strength and stiffness behavior in tension and compression are obtained. An algorithm for numerical solution of corresponding boundary-value problems is proposed and implemented. Results of numerical analysis were compared to acquired data for polymer matrix and structural carbon fiber reinforced plastics.

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